## Introductioin to the Riemann Integral

The Riemann integral is a method of integration which approximates the integral as lower and upper limits of a sum of terms. For the lower limit we split the region of integration up into a sequence of intervals, take the lowest value of on each interval, multiply this value by the length of the interval, and add the resulting terms. For the upper limit we take the largest value of on each interval, multiply this value by the length of the interval, and add the resulting terms. If the function is Riemann integrable, the the upper and lower limits are equal as the number of intervals tends to infinity and the length of each interval tends to zero. We must make this rigorous.

Let be an interval of and let be a bounded function. For any finite set of points such that there is a corresponding partition of Let be the set of all partitions of with Then let be the infimum of the set of upper Riemann sums with each partition in and let be the supremum of the set of lower Riemann sums with each partition in If then so is decreasing and is increasing. Moreover, and are bounded by Therefore, the limits and exist and are finite. If then is Riemann-integrable over and the Riemann integral of over is defined by Example: Show that is Riemann integrable over  is an increasing function. If we split the region of integration into n equal intervals each of length then each interval is except for the upper end interval which is In the interval and in the interval  therefore  Hence Let then the upper and lower limits are equal - - so is Riemann integrable over and  